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On the large genus asymptotics of Weil-Petersson volumes

Peter Zograf

TL;DR

The paper introduces a fast KdV-based algorithm to compute Weil-Petersson volumes and related ψ/κ intersection numbers on moduli spaces of curves. It then presents conjectured large-genus asymptotics for these volumes and intersection numbers, supported by extensive numerical data and consistent with known fixed-n asymptotics. The key findings include explicit asymptotic forms for V_{g,n} and V_{g,n;d}, along with ratio limits that appear universal across n, all of which await rigorous proofs. These results have potential implications for algebraic geometry, combinatorics, dynamical systems, and string theory, indicating both practical computational gains and deeper structural patterns in the geometry of moduli spaces.

Abstract

A relatively fast algorithm for evaluating Weil-Petersson volumes of moduli spaces of complex algebraic curves is proposed. On the basis of numerical data, a conjectural large genus asymptotics of the Weil-Petersson volumes is computed. Asymptotic formulas for the intersection numbers involving $ψ$-classes are conjectured as well. The accuracy of the formulas is high enough to believe that they are exact.

On the large genus asymptotics of Weil-Petersson volumes

TL;DR

The paper introduces a fast KdV-based algorithm to compute Weil-Petersson volumes and related ψ/κ intersection numbers on moduli spaces of curves. It then presents conjectured large-genus asymptotics for these volumes and intersection numbers, supported by extensive numerical data and consistent with known fixed-n asymptotics. The key findings include explicit asymptotic forms for V_{g,n} and V_{g,n;d}, along with ratio limits that appear universal across n, all of which await rigorous proofs. These results have potential implications for algebraic geometry, combinatorics, dynamical systems, and string theory, indicating both practical computational gains and deeper structural patterns in the geometry of moduli spaces.

Abstract

A relatively fast algorithm for evaluating Weil-Petersson volumes of moduli spaces of complex algebraic curves is proposed. On the basis of numerical data, a conjectural large genus asymptotics of the Weil-Petersson volumes is computed. Asymptotic formulas for the intersection numbers involving -classes are conjectured as well. The accuracy of the formulas is high enough to believe that they are exact.

Paper Structure

This paper contains 3 sections, 2 theorems, 20 equations.

Table of Contents

  1. Algorithms
  2. Asymptotics
  3. Numerics

Key Result

Theorem 1

Let where ($J_0$ denotes the Bessel function of the first kind). Then (i) the KdV equation has a unique solution $u(y,t)=y+h^{2g}\sum_{g=1}^\infty u_g(y,t)$ where each $u_g(y,t)$ is a Laurent polynomial in $t$ of the form (ii) for each $g\geq 2$ the equation has a unique solution of the form (iii) for any $g,n\geq 0$ the Weil-Petersson volume of $\mathcal{M}_{g,n}$ is given by the formula

Theorems & Definitions (4)

  • Theorem 1
  • Theorem 2
  • Conjecture 1
  • Conjecture 2